Eulerian integral associated with product of two multivariable I-functions,
a class of polynomials and the multivariable A-function
1 Teacher in High School , France E-mail : [email protected]
ABSTRACT
The present paper is evaluated a new Eulerian integral associated with the product of two multivariable I-functions defined by Nambisan et al [3] a generalized Lauricella function , a class of multivariable polynomials and multivariable A-function with general arguments . We will study the case concerning the multivariable H-function defined by Srivastava et al [10] and Srivastava-Daoust polynomial [6].
Keywords: Eulerian integral, multivariable I-function, generalized Lauricella function of several variables, multivariable A-function, generalized hypergeometric function, class of polynomials, Srivastava-Daoust polynomials
2010 Mathematics Subject Classification. 33C45, 33C60, 26D20
1. Introduction
In this paper, we evaluate a new Eulerian integral of most general characters associated with the products of two multivariable I-functions defined by Nambisan et al [3], the A-function of several variables defined by Gautam et al [2] and a class of polynomials with general arguments. The A-function of several variables is an extension of the multivariable H-function defined by Srivastava et al [10].
The serie representation of the multivariable A-function is given by Gautam [2] as
(1.1)
where
(1.2)
(1.3)
which is valid under the following conditions : (1.5)
and
(1.6)
Here ;
The I-function is defined and represented in the following manner.
=
(1.7)
(1.8)
where , , are given by :
(1.9)
(1.10)
For more details, see Nambisan et al [3].
Following the result of Braaksma [1] the I-function of r variables is analytic if
(1.11)
The integral (2.1) converges absolutely if
(1.12)
Consider the second multivariable I-function.
=
(1.13)
(1.14)
where , , are given by :
(1.15)
(1.16)
For more details, see Nambisan et al [3].
Following the result of Braaksma [1] the I-function of r variables is analytic if :
(1.17)
The integral (2.1) converges absolutely if
where
(1.18)
= (1.19)
The coefficients are arbitrary constants, real or complex.
2. Integral representation of generalized Lauricella function of several variables
The following generalized hypergeometric function in terms of multiple contour integrals is also required [8 ,page 39 eq .30]
(2.1)
where the contours are of Barnes type with indentations, if necessary, to ensure that the poles of
are separated from those of . The above result (1.23) can be easily established by an appeal to the calculus of residues by calculating the residues at the poles of
In order to evaluate a number of integrals of multivariable I-function, we first establish the formula
(2.2)
where
(2.3)
Here the contour are defined by starting at the point and terminating at the
point with and each of the remaining contour run from to
(2.2) can be easily established by expanding by means of the formula :
(2.4)
integrating term by term with the help of the integral given by Saigo and Saxena [4, page 93, eq.(3.2)] and applying the definition of the generalized Lauricella function [6, page 454].
3. Eulerian integral
In this section , we note :
;
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
; (3.8)
(3.9)
(3.14)
(3.15)
(3.16) j
(3.17) j
( (1.18)
(3.19)
(3.20)
(3.21)
where
for are defined respectively by (1.2) and (1.3)
(3.23)
(3.24)
(3.25) (3.25)
Provided that
(A ) ,
(B) ;
The exposants
of various gamma function involved in (1.10) and (1.11) may take non integer values.
The exposants
of various gamma function involved in (1.15) and (1.16) may take non integer values.
(C)
(D)
(E)
(F)
;
;
(G)
either and
or and
(I)
The multiple series occuring on the right-hand side of (3.25) is absolutely and uniformly convergent.Proof
To prove (3.25), first, we express in serie the multivariable A-function with the help of (1.1), a class of multivariable polynomials defined by Srivastava et al [7] in serie with the help of (1.19) and we interchange the order of summations and t-integral (which is permissible under the conditions stated). Expressing the I-functions of r-variables and s-r-variables defined by Nambisan et al [3] in terms of Mellin-Barnes type contour integral with the help of (1.8) and (1.14) respectivel and interchange the order of integrations which is justifiable due to absolute convergence of the integral involved in the process. Now collect the power of Now collect the power of with with
and collect the power of
and collect the power of with with . Use the equations (2.2). Use the equations (2.2) and (2.3) and express the result in Mellin-Barnes contour integral. Interpreting the
and (2.3) and express the result in Mellin-Barnes contour integral. Interpreting the dimensional dimensional Mellin-Barnes integral in multivariable I-function defined by Nambisan et al [3], we obtain the equation (3.25).
Mellin-Barnes integral in multivariable I-function defined by Nambisan et al [3], we obtain the equation (3.25).
Remarks
Remarks
If a)
If a) ; b) ; b) , we obtain the similar formulas that, we obtain the similar formulas that (3.25) with the corresponding simplifications.
(3.25) with the corresponding simplifications.
4.Particular case
a) , the multivariable I-functions defined by Nambisan
under the same conditions and notations that (3.25) with
b)
b) If If = = (4.2) (4.2)
then the general class of multivariable polynomial
then the general class of multivariable polynomial reduces to generalized Lauricella function reduces to generalized Lauricella function defined by Srivastava et al [6]. We have the following integral.
(4.3) (4.3)
under the same conditions and notations that (3.25) under the same conditions and notations that (3.25)
where
where , , is defined by (4.2) is defined by (4.2)
Remark:
By the following similar procedure, the results of this document can be extented to product of any finite number of By the following similar procedure, the results of this document can be extented to product of any finite number of multivariable I-functions defined by Nambisan et al [3] and a class of multivariable polynomials defined by Srivastava multivariable I-functions defined by Nambisan et al [3] and a class of multivariable polynomials defined by Srivastava et al [7].
et al [7].
5. Conclusion
In this paper we have evaluated a generalized Eulerian integral involving the product of two multivariable I-functions defined by Nambisan et al [3], a expansion of multivariable A-function and a class of multivariable polynomials defined by Srivastava et al [7] with general arguments. The formulae established in this paper is very general nature. Thus, the results established in this research work would serve as a key formula from which, upon specializing the parameters, as many as desired results involving the special functions of one and several variables can be obtained.
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